Transformation (function)

{{Short description|Function that applies a set to itself}} {{Redirect|Transformation (mathematics)||Transformation (disambiguation)}} {{broader|Function (mathematics)}} [[File:A code snippet for a rhombic repetitive pattern.svg|thumb|upright=1.5|A composition of four mappings coded in SVG, which '''transforms''' a rectangular repetitive pattern into a rhombic pattern. The four transformations are linear.]]

In mathematics, a '''transformation''', '''transform''', or '''self-map'''<ref>{{Cite web|title=Self-Map -- from Wolfram MathWorld|url=https://mathworld.wolfram.com/Self-Map.html|access-date=March 4, 2024}}</ref> is a function ''f'', usually with some geometrical underpinning, that maps a set ''X'' to itself, i.e. {{nowrap|''f'': ''X'' → ''X''}}.<ref>{{cite book|author1=Olexandr Ganyushkin|author2=Volodymyr Mazorchuk|title=Classical Finite Transformation Semigroups: An Introduction|url=https://archive.org/details/classicalfinitet00gany_719|url-access=limited|year=2008|publisher=Springer Science & Business Media|isbn=978-1-84800-281-4|page=[https://archive.org/details/classicalfinitet00gany_719/page/n73 1]}}</ref><ref name="Grillet1995">{{cite book|author=Pierre A. Grillet|title=Semigroups: An Introduction to the Structure Theory|url=https://books.google.com/books?id=yM544W1N2UUC&pg=PA2|year=1995|publisher=CRC Press|isbn=978-0-8247-9662-4|page=2}}</ref><ref>{{cite book|author=Wilkinson, Leland |title=The Grammar of Graphics|publisher=Springer|year=2005|isbn=978-0-387-24544-7|page=29|url=https://books.google.com/books?id=NRyGnjeNKJIC&pg=PA29|edition=2nd}}</ref> Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine transformations, such as rotations, reflections and translations.<ref>{{Cite web|url=https://www.mathsisfun.com/geometry/transformations.html|title=Transformations|website=www.mathsisfun.com|access-date=2019-12-13}}</ref><ref name=":0">{{Cite web|url=https://www.basic-mathematics.com/transformations-in-math.html|title=Types of Transformations in Math|website=Basic-mathematics.com|access-date=2019-12-13}}</ref>

== Partial transformations == While it is common to use the term '''transformation''' for any function of a set into itself (especially in terms like "transformation semigroup" and similar), there exists an alternative form of terminological convention in which the term "transformation" is reserved only for bijections. When such a narrow notion of transformation is generalized to partial functions, then a '''partial transformation''' is a function ''f'': ''A'' → ''B'', where both ''A'' and ''B'' are subsets of some set ''X''.<ref name="Hollings2014">{{cite book|author=Christopher Hollings|title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups|url=https://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA251|year=2014|publisher=American Mathematical Society|isbn=978-1-4704-1493-1|page=251}}</ref>

==Algebraic structures== The set of all transformations on a given base set, together with function composition, forms a regular semigroup.

==Combinatorics== For a finite set of cardinality ''n'', there are ''n''''n'' transformations and (''n''+1)''n'' partial transformations.<ref>{{cite book|author1=Olexandr Ganyushkin|author2=Volodymyr Mazorchuk|title=Classical Finite Transformation Semigroups: An Introduction|url=https://archive.org/details/classicalfinitet00gany_719|url-access=limited|year=2008|publisher=Springer Science & Business Media|isbn=978-1-84800-281-4|page=[https://archive.org/details/classicalfinitet00gany_719/page/n74 2]}}</ref>

==References== {{Reflist}}

==External links== *{{Commonscatinline}}

{{DEFAULTSORT:Transformation (Geometry)}} Category:Transformation (function) Category:Functions and mappings